View source for Rectified 6-simplexes

{| class=wikitable align=right width=450 style="margin-left:1em;"
|- align=center valign=top
|[[File:6-simplex t0.svg|150px]]<BR>6-simplex<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}
|[[File:6-simplex t1.svg|150px]]<BR>Rectified 6-simplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|3|node}}
|[[File:6-simplex t2.svg|150px]]<BR>Birectified 6-simplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node}}
|-
!colspan=3|[[Orthogonal projection]]s in A<sub>6</sub> [[Coxeter plane]]
|}
In six-dimensional [[geometry]], a '''rectified 6-simplex''' is a convex [[uniform 6-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[6-simplex]].

There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the ''rectified 6-simplex'' are located at the edge-centers of the ''6-simplex''. Vertices of the ''birectified 6-simplex'' are located in the triangular face centers of the ''6-simplex''.

== Rectified 6-simplex==
{{Uniform polypeton db|Uniform polypeton stat table|ril}}
[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|1|6}}. It is also called '''0<sub>4,1</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3a|nodea|3a|nodea|3a|nodea}}.

=== Alternate names ===
* Rectified heptapeton (Acronym: ril) (Jonathan Bowers)

=== Coordinates ===
The vertices of the ''rectified 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on [[Facet (geometry)|facets]] of the [[rectified 7-orthoplex]].

=== Images ===
{{6-simplex Coxeter plane graphs|t1|150}}

== Birectified 6-simplex==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Birectified 6-simplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|Class||[[A6 polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>2</sub>{3,3,3,3,3}<BR>2r{3<sup>5</sup>} = {3<sup>3,2</sup>}<BR>or <math>\left\{\begin{array}{l}3, 3, 3\\3, 3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter symbol]]|| 0<sub>32</sub>
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node}}<BR>{{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea}}
|-
|bgcolor=#e7dcc3|5-faces||14 total:<BR>7 [[Rectified 5-simplex|t<sub>1</sub>{3,3,3,3}]]<BR>7 [[Birectified 5-simplex|t<sub>2</sub>{3,3,3,3}]]
|-
|bgcolor=#e7dcc3|4-faces||84
|-
|bgcolor=#e7dcc3|Cells||245
|-
|bgcolor=#e7dcc3|Faces||350
|-
|bgcolor=#e7dcc3|Edges||210
|-
|bgcolor=#e7dcc3|Vertices||35
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||{3}x{3,3}
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Heptagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||A<sub>6</sub>, [3,3,3,3,3]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
[[Emanuel Lodewijk Elte|E. L. Elte]] identified it in 1912 as a semiregular polytope, labeling it as S{{supsub|2|6}}. It is also called '''0<sub>3,2</sub>''' for its branching Coxeter-Dynkin diagram, shown as {{CDD|node_1|split1|nodes|3ab|nodes|3a|nodea}}.

=== Alternate names ===
* Birectified heptapeton (Acronym: bril) (Jonathan Bowers)

=== Coordinates ===
The vertices of the ''birectified 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on [[Facet (geometry)|facets]] of the [[birectified 7-orthoplex]].

=== Images ===
{{6-simplex Coxeter plane graphs|t2|150}}

== Related uniform 6-polytopes ==

The rectified 6-simplex polytope is the [[vertex figure]] of the [[7-demicube]], and the [[edge figure]] of the uniform [[2 41 polytope|2<sub>41</sub> polytope]].

These polytopes are a part of 35 [[Uniform 6-polytope#The A6 .5B3.2C3.2C3.2C3.2C3.5D family .286-simplex.29|uniform 6-polytopes]] based on the [3,3,3,3,3] [[Coxeter group]], all shown here in A<sub>6</sub> [[Coxeter plane]] [[orthographic projection]]s.

{{Heptapeton family}}

== Notes==
{{reflist}}

== References ==
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: 
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 
** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{isbn|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. 
*{{KlitzingPolytopes|polypeta.htm|6D|uniform polytopes (polypeta)}} o3x3o3o3o3o - ril, o3x3o3o3o3o - bril

== External links ==
* [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

[[Category:6-polytopes]]